ISOPERIMETRY. 



Prop. XV, 



A cylinder has a lefs furface than any prifm of equal bafe 

 «:id altitude. 



For fince the bafes and altitudes are equal by hypothefis, 

 the furfaces will be greater or lefs, as the lateral furfaces are 

 greater or lefi: ; but thefe will be as the perimeter!, of the 

 (olid bafes, of which that of the cyhndcr will be the Icail, 

 being a circle (by Prop. X.), and confequently the cyhnder 

 is tliat which has the lead furface. 



Cor. I — And again converfely, of all folids on equal bafes, 

 and whofe furfaces are alfo equal, the cylinder is that which 

 has the greatell capacity. 



Cor. 2. -In the fame manner (by Prop. VI. Cor. i.) it 

 n.ay be demonilrated, that of all right prifms of the fame alti- 

 tude, and whofe bafes are equal, and of the fame number of 

 fide« ; that has the lead furface whofe bafe is a regular 

 figure, and therefore when the prifm is a parallelepiped the 

 bafe is a fquare. 



Cor. ^. — And again, of all right prifms, whofe altitudes 

 and furfaces are equal, and whofe bafes have a given number 

 «f fides, that which has a regular figure for its bafe is the 

 grea'elt ; and therefore when the priim is a parallelopiped 

 the bafe is a fquare. 



Prop. XVI. 



Of all right parallelepipeds, given in magnitude, that 

 Vfhich has the fmal'ell furface has all iis faces equal, or is a 

 cube : and reciprocally of all parallelopipcds of equal fur- 

 face, the greateil is a cube. 



For by the foregoing corollaries, the right parallelopiped, 

 having the fmalleil f.:rl'ace, with the fame capacity, or the 

 greatell capacity with the fame furface has a fquare for 

 its bafe : but any face whatever may be taken for the 

 bafe : therefore in the parallelopiped, whofe furface is the 

 Imalleft with the fame capacity, or whofe capacity is the 

 greateft with the fame furface, has necelfarily every two of 

 its oppofite faces fquares, and confequently it is a cube. 

 Q. E. D. 



Prop XVII. 



Of all cylinders of the fame capacity, that has the leaft 

 furface whofe altitude is equal to the diameter of its bafe. 

 Fi-,s. 12. and ij. 



Let A B C D, and abed, be two cylinders of equal ca- 

 pacity, and of which the firft, A B C D, has its ahitude 

 ?qual to the diameter of its bafe, and the other any cy- 

 linder who.fe dimenfions are not the fame with the firlt, then 

 I fay the cylinder A B C D has the leaft furface. 



For conceive each of thefe cyHnders to be circumfcribed 

 by a fquare prifm, then will the capacities of thefe prifms be 

 alfo equal ; and their furfaces will be to each other as the 

 furfaces of the cylinders that they circumfcribe, as is evident 

 from (Cor. i. Prop. IX.) ; and therefore reciprocally, the 

 farfaces of the cyhnders will be to each other as the fur- 

 faces of the prifms : but fince thefe prifms have equal capa- 

 cities, that which circumfcribes the cyhnder A B C D has 

 the leaft furface, becaule it is a cube (Prop. XVI.) ; and 

 confequently, the furface of that cylinder is the leaft alfo. 

 Q. li D. 



Cor. — In a fimilar manner it is demonftrated, that of all 

 cylinders of the fame furface, that has the greateft fohdity 

 whofe altitude is equal to the diameter of its bafe. 



Ifoperinutrical Prchkms — We have before obferved, that 

 the theorems relating to the furfaces and foiidities of bodies, 

 of equal perimeters, might be confidered as forming the ele- 

 ments of if ^perimetry, and which, as we have feen, are de- 



monftrable from the fimple elements of geometry; wl'le 

 tliofe relating to the maxima et minima of curves are of th- 

 highell order of problems ; which have called ip'o aftion tic 

 talents, and excited the pafTions, of foine of the ableft geo- 

 meters of modern tunes, having led to a difpute, whicti, 

 for want of impartial and competent judges, remained uiicii- 

 cided for many years, and which has fince been termed " C. •. 

 war of problems," on account of the great intereft it t,v- 

 cited, and the determined and able nianner in which eacli 

 party fupported its opinion, and contelled that of its oppo- 

 nent ; and as this difpute cannot but be confidered as one 

 of the moil memorable ever s in the hillory of the modern 

 analyfis, we (hall prefent the .xader with an abllradt of it, 

 fo far as it relates to iloperimetry, referring him for furtl. i 

 information to the " Hiiloire des Mathematiques," by 

 Montucla, vol. iii. p. 322, ar.d to BofTut's "Hillory of 

 Mathematics," p. 331 ; and alfo to an interciling little 

 treatife on this fubjeCl, lately publifhed by Mr. Woodhoufe 

 of Caius college, Cambridge. 



The firft problem which can be faid to relate to this clafs, 

 was propofed by Newton in liis " Principia,"' which was that 

 of the folid of leaft refiftance. But the fubjcdt and doftrine 

 did not become a matter of difcuffion and controverfy, till 

 John Bernouilli required of mathematicians the determination 

 of the curve of quickeft defcent, in a paper publifhed in the 

 Leipfic ads for June 1696, under ths following form : 



Problema Novum 

 ad cnjus folutionem mathematici invitantur. 

 " Datis in piano verticali duobus punctis A et B, aflignare 

 mobili M, viam A M B, per quam gravitate fu3 defcendens, 

 et moveri incipiens a puncto A, breviflimo tempore perveniat 

 ad altrum punctum B." /'y' H- 



At the firll view of this problem, it would be imagined, that 

 a right line, as it is the ihorteft path from one point to an- 

 other, muft likewife be the line of fwifteft defcent : but the 

 attentive geometer will not haftily alTert this, when he con- 

 fiders that in a concave curve deicnbcd from one point to 

 another, the moving body defcends ai firft in a direction more 

 approaching to a perpendicular, and confequently acquires 

 a greater velocity than down an inclined plane ; which 

 greater velocity is to be fet again il the length of the path, 

 which may caufe the body to arrive at the point B fooner 

 through the curve than down the plane. Metaphyfics alone, 

 therefore, cannot folve the quettion ; in fad it requires the 

 utmoft accuracy of mathematical inveftigation and calcula- 

 tion, the refulc of which (hews that the path required is a cy- 

 cloid reverfed, as we (hall fee in what follows, being at 

 that time a new and remarkable property of this curve-, which 

 the refearches of Huygens and Palcal had prevloufly ren- 

 dered fo celebrated. See Cycloid. 



According to BoiTat, Leibnitz rcfolved this problem the 

 fame day on which he received it, but that he and John Ber- 

 nouilli agreed to keep back their folutions ; but the fad of 

 Leibnitz having obtained a correft folution ftcins to be 

 very doubtful ; at all events, at the e.-ipiration of fix months, 

 the time allowed, no folution was publiihed, and the time 

 was accordingly enlarged to one year, during which period 

 the folutions of James BcrnouiUi, Newton, and the marquis 

 de I'Hopital appeared ; Bernouilli's and Newton's were 

 both given in the Ada. Erud. Lipf for May 1697, but the 

 latter without a name, the real author of wliicli, however, 

 mathematicians had little trouble in divining, for as John 

 Bernouilli obferved on this occafion, " ex ungue leoncm." 



James Bernouilli, in the courfe of his inveftigations, had 



afcended to problems on ifoperimetrical figures requiring ftlll 



more profound fpeculations, and, (ifttr having rcfolved 



5 tliem* 



